Dec 1 Relativity and Gravity

General relativity could be regarded as being one of the greatest achievements of science in human history. As early as the 1600’s Isaac Newton was considering the causes and effects of gravity. He formulated his famous equation of the gravitational force;

Figure 1 - Newtonian Gravity – Wikipedia

It states that the force between two bodies is inversely proportion to the square of the distance between them. This Newtonian theory of gravity stood the test of time for centuries until the early 1900’s when Albert Einstein constructed his new and radical approach to gravity – a geometric theory of gravity.

Einstein’s geometric theory moved away from bodies experiencing a ‘force’ and explained gravity as being an effect of the shape of space-time, which can be imagined as a flexible elastic sheet.

Objects like the Earth deform the fabric of space-time a little, whereas heavy objects like stars and galaxies deform it greatly.

This theory is expressed in a compact and beautiful way;

Figure 2 - Einstein Field Equation

The left hand side of the equation describes the geometry of space-time, and the right hand side describes the energy of that area of space. The old adage;

“Matter tells space how to curve, space tells matter how to move”

These equations are notoriously difficult to solve, each component with subscript µν will have 256 components if dealing with 4 dimensions. However, one of the first people to find a physically interesting solution to the Einstein field equation was Karl Schwarzschild, while he was serving on the front line during World War I. He made some simplifying assumptions about the nature of space-time which made finding a solution far easier, and what he ended up with, is now named after him;

Figure 3 - Schwarzschild Line Element

Though this equation looks far more complicated than the previous by Einstein, this equation represents the “line element” (i.e. Pythagoras Theorem) in Schwarzschild space. It actually produced some really interesting physics, including one of the first mathematical descriptions of a black hole. As mentioned above, Schwarzschild made some assumptions about his black hole model;

1. Space was asytotically flat (at large distances from the mass, space was flat)

2. The object was stationary (not rotating)

3. The object was static (not changing in time)

By making these assumptions Schwarzschild was able to formulate his solution to the field equation, and describe a type of black hole which now takes his name. It doesn’t represent black holes as we know them no, as it does not take into account the rotation (angular momentum) but it did allow scientists to explore gravitational redshift, gravitational time dilation, gravitational tidal forces and much more. DON’T FORGET Karl Schwarzschild was fighting a war when he did this work!!

Another interesting equation to pop out of the Schwarzschild equation is what’s known as the Schwarzschild radius (Rs). If an object were to be compressed below its Schwarzschild radius, it would continue to collapse, and form a black hole.

Figure 4 - Schwarzschild Radius Equation

For our sun, Rs is about 3km (and our sun has a radius of about 700,000km!) For the Earth, Rs is about 9mm!!!

The Schwarzschild solution formed the basis for most of the other solutions, including the Kerr solution, which builds on that of Schwarzschild, but introduces angular momentum (spin) into the equation;

Figure 5 - Kerr Solution

The Kerr solution is a more realistic description of black holes, due to this requirement to include angular momentum. It is a conserved quantity in physics, and we observe stars to rotate, therefore as the star contracts to form a black hole, it would “spin-up” (Pulsars are a perfect example of this spin-up process, some having rotational periods of milliseconds)

Rotation causes another odd but fascinating effect – Frame Dragging. The rotation of the black hole, literally starts to pull space around it, like the water of your bath going down the plug hole. Space begins to whirl around the black hole.